The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. The construction first establishes the circumcenter and then draws the circle. circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. This page shows how to construct (draw) the circumcircle of a triangle with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

The image below is the final drawing above with the red labels added.

Note: This proof is almost identical to the proof in Constructing the circumcenter of a triangle.

Argument | Reason | |
---|---|---|

1 | JK is the perpendicular bisector of AB. | By construction. For proof see Constructing the perpendicular bisector of a line segment |

2 | Circles exist whose center lies on the line JK and of which AB is a chord. (* see note below) | The perpendicular bisector of a chord always passes through the circle's center. |

3 | LM is the perpendicular bisector of BC. | By construction. For proof see Constructing the perpendicular bisector of a line segment |

4 | Circles exist whose center lies on the line LM and of which BC is a chord. (* see note below) | The perpendicular bisector of a chord always passes through the circle's center. |

5 | The point O is the circumcenter of the triangle ABC, the center of the only circle that passes through A,B,C. | O is the only point that lies on both JK and LM, and so satisfies both 2 and 4 above. |

5 | The circle O is the circumcircle of the triangle ABC. | The circle passes through all three vertices A, B, C |

- Q.E.D

*** Note **

Depending where the center point lies on the bisector, there is an infinite number of circles that can satisfy this.
Two of them are shown below.
Steps 2 and 4 work together to reduce the possible number to just one.

- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)

- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more

- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)

- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)

- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle

- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle

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